\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 138, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2011/138\hfil Sublinear Hamiltonian systems]
{Subharmonic solutions for non-autonomous second-order sublinear
Hamiltonian systems with $p$-Laplacian}
\author[Z. Wang\hfil EJDE-2011/138\hfilneg]
{Zhiyong Wang}
\address{Zhiyong Wang \newline
Department of Mathematics,
Nanjing University of Information Science \& Technology,
Nanjing 210044, Jiangsu, China}
\email{mathswzhy1979@gmail.com}
\thanks{Submitted August 1, 2011. Published October 20, 2011.}
\subjclass[2000]{34B15, 34K13, 58E30}
\keywords{Control functions; subharmonic solutions;
$p$-Laplacian systems; \hfill\break\indent minimax methods}
\begin{abstract}
In this article, we study the existence of subharmonic
solutions to the non-autonomous second-order sublinear Hamiltonian
systems with $p$-Laplacian. Introducing some kinds of control
functions, infinitely many subharmonic solutions are obtained by
using the minimax methods in critical point theory. We point out
that our results are new even in the case $p=2$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction and main results}
Consider the second-order system
\begin{equation} \label{1.1}
\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))+\nabla F(t,u(t))=0\quad
\text{a.e. }t\in\mathbb{R}.
\end{equation}
where $p>1$, $F:\mathbb{R}\times \mathbb{R^N}\to\mathbb{R}$ is $T$-periodic ($T>0$)
in its first variable for all $x\in \mathbb{R^N}$, and satisfies the
assumption
\begin{itemize}
\item[(A1)] $F(t,x)$ is measurable in $t$ for every
$x\in \mathbb{R^N}$ and continuously differentiable in $x$ for a.e.
$t\in[0,T]$, and there exist $a\in C(\mathbb{R^+},\mathbb{R^+}),b\in
L^1(0,T;\mathbb{R^+})$ such that
\[
|F(t,x)|+|\nabla F(t,x)|\leq a(|x|)b(t)
\]
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$.
\end{itemize}
A solution is called subharmonic solution if it is
$kT$-periodic solution for some positive integer $k$
(see for example \cite{Mawhin89}).
Recently, considerable attention has been paid to
subharmonic solutions of sec\-ond-order Hamiltonian systems with
$p$-Laplacian; see \cite{Xu07, Pasca08, Mawhin00, Jebelean07,
Tian07, Zhang09, Wang09}. When $p=2$, Equation \eqref{1.1} reduces
to the second-order non-autonomous Hamiltonian system
\begin{equation} \label{1.2}
\ddot{u}(t)+\nabla F(t,u(t))=0\quad \text{a.e. }t\in\mathbb{R}.
\end{equation}
Using the variational methods, many existence results are obtained
under suitable conditions, we refer the reader to \cite{Daouas03,
Tang98, Tang01, Tang02, Tang05, Zhao04, Ekeland02, Mawhin89,
Schechter06, Berger77, Cac96, Hirano98, Rabinowitz80, Felmer93,
Jiang07, Wang10} and the reference therein.
In particular, in \cite{Tang05}, Tang and Wu have proved the
following result.
\begin{theorem}[\cite{Tang05}] \label{thmA}
Suppose that $F$ satisfies assumption {\rm (A1)}
and the following conditions:
\begin{itemize}
\item[(S1)] There exist $f,g\in L^1(0,T;\mathbb{R^+})$ and
$\alpha\in[0,1)$ such that
\[
|\nabla F(t,x)|\leq f(t)|x|^\alpha+g(t)
\]
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$;
\item[(S2)] There exists $\gamma\in L^1(0,T;\mathbb{R})$ such that
\[
F(t,x)\geq \gamma(t)
\]
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$;
\item[(S3)] There exists a subset $E$ of $[0,T]$ with $\operatorname{meas}(E)>0$
such that
\[
\frac{1}{|x|^{2\alpha}}F(t,x)\to+\infty\quad \text{as }
|x|\to +\infty
\]
for a.e. $t\in E$.
\end{itemize}
Then problem \eqref{1.2} has a $kT$-periodic solution $u_k$ for every
positive integer $k$, and
$\max_{0\leq t\leq kT}|u(t)|\to+\infty$ as $k\to+\infty$.
\end{theorem}
Subsequently, Pasca and Tang in \cite{Pasca08} dealt with
the second order differential inclusions systems with $p$-Laplacian.
They generalized Theorem \ref{thmA} in a more general sense.
Note that in \cite{Tang05, Pasca08},
it is usually assumed that
(S1) holds, for $p$-Laplacian systems, $\alpha\in[0,p-1)$.
This means that nonlinearity $\nabla F(t,x)$ is sublinear.
Recently, the author and Zhang \cite{Wang10}, introduced a
control function $h(t)$, consider the case in which nonlinearity
$\nabla F(t,x)$ is only weak sublinear:
It is assumed that there exists a positive function
$h\in C(\mathbb{R^+},\mathbb{R^+})$ satisfied the following
restrictions
\begin{itemize}
\item[(i)] $h(s)\leq h(t)$ for all $s\leq t$, $s,t\in \mathbb{R^+}$;
\item[(ii)] $h(s+t)\leq C^*(h(s)+h(t))$ for all $s,t\in \mathbb{R^+}$;
\item[(iii)] $ 0< h(t)\leq K_1t^\alpha +K_2$ for all
$t\in \mathbb{R^+}$;
\item[(iv)] $h(t)\to+\infty$ as $t\to+\infty$.
\end{itemize}
Here $C^*,K_1,K_2$ are positive constants, $\alpha\in[0,1)$, and
$h(t)$ need just to satisfy conditions (i)-(iii) if $\alpha=0$.
Moreover, conditions
\begin{gather*}
|\nabla F(t,x)|\leq f(t)h(|x|)+g(t),\\
\frac{1}{h^2(|x|)}\int_0^TF(t,x)dx\to\pm\infty\quad
\text{as }|x|\to+\infty
\end{gather*}
are posed. Under these assumptions, they show that second-order
system
\begin{equation} \label{1.3}
\begin{gathered}
\ddot{u}(t)=\nabla F(t,u(t))\quad \text{a.e. }t\in[0,T],\\
u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0
\end{gathered}
\end{equation}
has a $T$-periodic solution. In addition, if the nonlinearity
$\nabla F(t,x)$ grows slightly slower than $|x|^{p-1}$ at infinity,
such as
\begin{equation}\label{1.4}
\nabla F(t,x)=\frac{t|x|^{p-1}}{\ln(e+|x|^2)},
\end{equation}
solutions are saddle points of problem
\begin{equation}\label{1.5}
\begin{gathered}
\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))=\nabla F(t,u(t))\quad
\text{a.e. }t\in[0,T],\\
u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,
\end{gathered}
\end{equation}
which have been obtained in \cite{Wang09} by minimax methods.
In the present article, we will focus on the subharmonic solutions
for \eqref{1.1} by replacing in assumptions (S1) and (S3)
the term $|x|$ with more general control functions
$h(|x|)$. Here, we emphasize that our results are still new when
$p=2$.
We will establish our main results:
\begin{theorem} \label{thm1.1}
Suppose that $F$ satisfies assumption {\rm (A1)} and the following
conditions:
\begin{itemize}
\item[(H1)] There exist constants $C^*>0$, $K_1>0$, $K_2>0$,
$\alpha\in[0,p-1)$ and a positive function
$h\in C(\mathbb{R^+},\mathbb{R^+})$ with the
properties {\rm (i)--(iv)}. Moreover, there exist
$f,g\in L^1(0,T;\mathbb{R^+})$ such that
\[
|\nabla F(t,x)|\leq f(t)h(|x|)+g(t)
\]
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$;
\item[(H2)] There exists $\gamma\in L^1(0,T;\mathbb{R})$ such that
\[
F(t,x)\geq \gamma(t)
\]
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$;
\item[(H3)] There exist a positive function
$h\in C(\mathbb{R^+},\mathbb{R^+})$
which satisfies the conditions {\rm (i)--(iv)}, and a
subset $E$ of $[0,T]$ with $\operatorname{meas}(E)>0$ such that
\[
\frac{1}{h^q(|x|)}F(t,x)\to+\infty\quad\text{as }|x|\to+\infty
\]
for a.e. $t\in E$, here $q:=\frac{p}{p-1}$.
\end{itemize}
Then \eqref{1.1} has $kT$-periodic solution $u_k\in
W_{kT}^{1,p} $ for every positive integer $k$ such that
$\|u_k\|_\infty\to+\infty$ as $k\to+\infty$, where
\begin{align*}
W_{kT}^{1,p} :=\big\{&u:[0,kT]\to \mathbb{R^N}|\; u\text{ is
absolutely continuous},\\
& u(0)=u(kT),\dot{u}\in
L^p(0,kT;\mathbb{R^N})\big\}
\end{align*}
is a Banach space with the norm
\[
\|u\|:=\Big(\int_0^{kT}|u(t)|^pdt+\int_0^{kT}|\dot{u}(t)|^pdt\Big)^{1/p}
\]
and $\|u_k\|_\infty:=\max_{0\leq t\leq kT}|u(t)|$ for $u\in
W_{kT}^{1,p}$.
\end{theorem}
\begin{remark} \label{rmk1.1}\rm
Theorem \ref{thm1.1} generalizes
Theorem \ref{thmA}. In fact, when $p=2$, Theorem \ref{thmA}
is a special case of Theorem \ref{thm1.1} with control function
$h(t)=t^\alpha$, $\alpha\in[0,p-1)$, $t\in\mathbb{R^+}$.
Furthermore, there are functions $F(t,x)$ satisfying Theorem
\ref{thm1.1} and not satisfying Theorem \ref{thmA} and earlier
results in the references \cite{Daouas03, Xu07,
Tang98, Tang01, Tang02, Pasca08, Zhao04, Ekeland02, Mawhin89,
Mawhin00, Schechter06, Berger77, Cac96, Hirano98, Rabinowitz80,
Felmer93, Jebelean07, Jiang07, Tian07, Zhang09, Wang10, Wang09}, even
for $p=2$.
\end{remark}
\begin{example} \label{examp1.1} \rm
Consider the function
\[
F(t,x)=\sin[(1+|x|^2)^{1/2}\ln^{1/2}(e+|x|^2)]
+|\sin\omega t|\ln^{\frac{3}{2}}(e+|x|^2)
\]
for all $x\in \mathbb{R^N}$ and $t\in \mathbb{R}$, where
$\omega=2\pi/T$. It is apparent that
\[
|\nabla F(t,x)|\leq \ln^{1/2}(e+|x|^2)+10,
\]
which implies that $F(t,x)$ is not bounded. Moreover, one has
\[
\frac{1}{|x|^{2\alpha}}F(t,x)\to0\quad \text{as
}|x|\to+\infty
\]
for any $\alpha\in (0,1)$ and $t\in \mathbb{R}$. Hence, this example
can not be solved by Theorem \ref{thmA} and the results
in \cite{Daouas03,
Xu07, Tang98, Tang01, Tang02, Pasca08, Zhao04, Ekeland02, Mawhin89,
Mawhin00, Schechter06, Berger77, Cac96, Hirano98, Rabinowitz80,
Felmer93, Jebelean07, Jiang07, Tian07, Zhang09, Wang10, Wang09}.
On the other hand, put $h(t)=\ln^{1/2}(e+t^2)$, it is not
difficult to see that the properties (i)-(iv) of $h(t)$ are all
satisfied. By simple computation, (H1)-(H3) remains true.
Therefore, $F(t,x)$ satisfies all the conditions of Theorem
\ref{thm1.1}, then problem \eqref{1.1} with $p=2$ has
$kT$-periodic solution $u_k\in W_{kT}^{1,p} $ for every positive
integer $k$ such that $\|u_k\|_\infty\to+\infty$ as
$k\to+\infty$.
\end{example}
\begin{theorem} \label{thm1.2}
Suppose that $F$ satisfies assumption {\rm (A1), (H2)} and the
following conditions:
\begin{itemize}
\item[(H4)] There exists constant $C^*>0$
and a positive function $h^*\in C(\mathbb{R^+},\mathbb{R^+})$ with
the properties:
\begin{itemize}
\item[(i*)] $h^*(s)\leq h^*(t)$ for all $s\leq t$,
$s,t\in \mathbb{R^+}$;
\item[(ii*)] $h^*(s+t)\leq C^*(h^*(s)+h^*(t))$ for all
$ s,t\in \mathbb{R^+}$;
\item[(ii*)] $th^*(t)-pH^*(t)\to-\infty$ as $t\to+\infty$,
where $H^*(t)=\int_0^th^*(s)ds$;
\item[(iv*)] $H^*(t)/t^p \to 0$ as $t\to+\infty$.
\end{itemize}
Moreover, there exist $f,g\in L^1(0,T;\mathbb{R^+})$ such that
\[
|\nabla F(t,x)|\leq f(t)h^*(|x|)+g(t)
\]
for all $x\in \mathbb{R^N}$ and a.e. $t\in[0,T]$;
\item[(H5)] There exist a positive function
$h^*\in C(\mathbb{R^+},\mathbb{R^+})$
which satisfies the conditions {\rm (i*)--(iv*)}, and a
subset $E$ of $[0,T]$ with $\operatorname{meas}(E)>0$ such that
\[
\frac{1}{H^*(|x|)}F(t,x)>0\quad\text{as }|x|\to+\infty
\]
for a.e. $t\in E$.
\end{itemize}
Then \eqref{1.1} has $kT$-periodic solution $u_k\in
W_{kT}^{1,p} $ for every positive integer $k$ such that
$\|u_k\|_\infty\to+\infty$ as $k\to+\infty$.
\end{theorem}
\begin{remark} \label{rmk1.2} \rm
(1) In contrast to the result in Theorem \ref{thm1.1},
if $\nabla F(t,x)$ grows faster at infinity, with the rate like
$\frac{|x|^{p-1}}{\ln(e+|x|^2)}$, from the proof we see that,
the approach of Theorem \ref{thm1.1} can not be repeated
unless $f(t)$ satisfies certain restrictions, and $\alpha$
has a wider range, say, $\alpha\in[0,p-1]$.
Meanwhile Theorem \ref{thm1.2} needs only
$f(t)$ belonging to $L^1(0,T;\mathbb{R^+})$.
(2) Comparing with \cite{Wang09}, we emphasize that Theorem
\ref{thm1.2} can not only treat the case like \eqref{1.4},
but also cases like (S1)-(S3).
Details for this assertion can be found in Example \ref{examp1.2}
below, also in the example in Section 4.
Furthermore, our methods here are
simpler and more direct than those in \cite{Wang09}.
(3) We must point out that assumption (H4) leads to
$H^*(t)\to+\infty$ as $t\to+\infty$ (for details see
Lemma \ref{lemma2.2}), then (H5) is stronger than
(H3) (or (S3)) with $\alpha=0$. Therefore, Theorem
\ref{thm1.2} is a new result, and do not cover Theorem
\ref{1.1}.
(4) There are functions $F(t,x)$ satisfying Theorem
\ref{thm1.2} and not satisfying Theorem \ref{thmA},
Theorem \ref{thm1.1}, or the assumptions in
\cite{Daouas03, Xu07, Tang98, Tang01, Tang02,
Pasca08, Zhao04, Ekeland02, Mawhin89, Mawhin00, Schechter06,
Berger77, Cac96, Hirano98, Rabinowitz80, Felmer93, Jebelean07,
Jiang07, Tian07, Zhang09, Wang10, Wang09}.
\end{remark}
\begin{example} \label{examp1.2}\rm
Consider the function
\[
F(t,x)=|\sin\omega t||f(t)|\frac{|x|^p}{\ln(e+|x|^2)},
\]
where $f(t)\in L^1(\mathbb{R}, \mathbb{R^+})$. Clearly, for all
$x\in\mathbb{R^N}$ and $t\in \mathbb{R}$, one has
\[
|\nabla F(t,x)|\leq(2+p)|f(t)|\frac{|x|^{p-1}}{\ln(e+|x|^2)},
\]
which implies that (H1) does not hold for any
$\alpha\in [0,p-1)$. Moreover, as mentioned before $f(t)$
only belongs to $L^1(\mathbb{R},\mathbb{R^+})$ and no other
further requirements on $f(t)$ are posed, then the approach
of Theorem \ref{thm1.1} can not be applied.
This is the key feature that Theorem \ref{thm1.2} is
different from Theorem \ref{thm1.1}. Thus,
this example can not be solved by earlier results even if $p=2$.
On the other hand, take $h^*(t)=\frac{t^{p-1}}{\ln(e+t^2)}$,
$H^*(t)=\int_0^t\frac{s^{p-1}}{\ln(e+s^2)}ds,s,t\in\mathbb{R^+}$,
then we can find that conditions (H2), (H4) and (H5) are
all satisfied, by Theorem \ref{thm1.2}, problem \eqref{1.1} has
$kT$-periodic solution $u_k\in W_{kT}^{1,p} $ for every positive
integer $k$ such that $\|u_k\|_\infty\to+\infty$ as
$k\to+\infty$.
\end{example}
\begin{remark} \label{rmk1.3}\rm
Without loss of generality, we may
assume that functions $b$ in assumption (A1), $f,g$ in (H1),
(H4) and $\gamma$ in (H2) are $T$-periodic.
Then assumptions (A1), (H1), (H2), (H4) hold for all $t\in R$ by the
$T$-periodicity of $F(t,x)$ in the first variable.
\end{remark}
The remainder of this article is organized as follows.
In Section 2 we give some notations and the estimates of control
functions $h^*(t)$ and $H^*(t)$. Section 3 are devoted to the proofs
of main theorems.
Finally, we will give a new example to illustrate our results in
Section 4.
\section{Preliminaries}
Let $k$ be a positive integer. For convenience, in the following we
will denote various positive constants as $C_i,i=0,1,2,\cdots$. For
$u\in W_{kT}^{1,p}$, let $\bar{u}:=\frac{1}{kT}\int_0^{kT}u(t)dt$
and $\tilde{u}(t):=u(t)-\bar{u}$, then one has:
Sobolev's inequality
\[
\|\tilde{u}\|_\infty\leq
C_0\int_0^{kT}|\dot{u}(t)|^pdt
\]
and Wirtinger's inequality
\[
\int_0^{kT}|\tilde{u}(t)|^pdt\leq C_0\int_0^{kT}|\dot{u}(t)|^pdt.
\]
It follows from assumption (A1) that functional $\varphi_k$ on
$W_{kT}^{1,p}$ give by
\[
\varphi_k(u)=\frac{1}{p}\int_0^{kT}|\dot{u}(t)|^pdt-\int_0^{kT}F(t,u(t))dt
\]
is continuously differentiable on $W_{kT}^{1,p}$ (see
\cite{Mawhin89}). Moreover, one has
\[
(\varphi'_k(u),v)=\int_0^{kT}
(|\dot{u}(t)|^{p-2}\dot{u}(t),\dot{v}(t))dt -\int_0^{kT}(\nabla
F(t,u(t)),v(t))dt
\]
for all $u,v\in W_{kT}^{1,p}$. It is well known that the solutions
to problem \eqref{1.1} correspond to the critical points of the
functional $\varphi_k$.
To proof of our main theorems, we need the following auxiliary
results.
\begin{lemma}[\cite{Tang01}] \label{lemma2.1}
Suppose that $F$ satisfies assumption {\rm (A1)},
and $E$ is a measurable subset of $[0,T]$. Assume that
\[
F(t,x)\to+\infty\quad\text{as }|x|\to\infty
\]
for a.e. $t\in E$. Then for every $\delta>0$, there exists subset
$E_\delta$ of $E$ with
$\operatorname{meas}(E\backslash E_\delta)0$, $t\in \mathbb{R^+}$,
\item[(b)] ${h^*}^q(t)/ H^*(t) \to 0$ as $t\to+\infty$,
\item[(c)] $H^*(t)\to+\infty$ as $t\to+\infty$.
\end{itemize}
\end{lemma}
\begin{proof}
It follows from (iv*) that, for any
$\epsilon>0$, there exists $M_1>0$ such that
\[
H^*(t)\leq\varepsilon t^p\quad \forall t\geq M_1.
\]
Observe that by (iii*), there exists $M_2>0$ such
that
\[
th^*(t)-pH^*(t)\leq0\quad \forall t\geq M_2,
\]
which implies that
\[
h^*(t)\leq\frac{pH^*(t)}{t}\leq p\epsilon t^{p-1}\quad \forall t\geq
M,
\]
where $M:=\max\{M_1,M_2\}$. Hence we obtain
\[
h^*(t)\leq p\epsilon t^{p-1}+h^*(M)
\]
for all $t>0$ by ($\rm i^*$) of (H4). Obviously, $h^*(t)$
satisfies (a) due to the definition of $h^*(t)$ and the above
inequality.
Next, we turn to (b). Recalling property (iv*)
and the fact $q=\frac{p}{p-1}$, we obtain
\begin{align*}
00$, there exists $M_3>0$ such that
\[
th^*(t)-pH^*(t)\leq-L\quad \forall t\geq M_3.
\]
So, one has
\[
\theta th^*(\theta t)-pH^*(\theta t)\leq-L
\]
for all $|\theta t|\geq M_3$. Then we have
\[
\frac{d}{d\theta}\big[\frac{H^*(\theta
t)}{\theta^p}\big]
=\frac{\theta t\cdot h^*(\theta t)-pH^*(\theta
t)}{\theta^{p+1}}
\leq-\frac{L}{\theta^{p+1}}
=\frac{d}{d\theta}\big(\frac{L}{p\theta^p}\big).
\]
Let $\theta>1$, integrating both sides of the above inequality from
1 to $\theta$, we obtain
\[
\frac{H^*(\theta
t)}{\theta^p}-H^*(t)\leq\frac{L}{p\theta^p}-\frac{L}{p}
=\frac{L}{p}\big(\frac{1}{\theta^p-1}\big).
\]
Let $\theta\to+\infty$ in the above inequality, and by
(iv*), one has
\[
H^*(t)\geq\frac{L}{p}
\]
for all $t\geq M_3$. By the arbitrariness of $L$, we have
$H^*(t)\to+\infty$ as $t\to+\infty$.
This completes the proof.
\end{proof}
\section{Proof of main results}
In this section, firstly, we discuss the (PS) condition.
\begin{lemma} \label{lemma3.1}
Assume that $F$ satisfies assumptions {\rm (A1), (H1)--(H3)}. Then
$\varphi_k$ satisfies the {\rm (PS)} condition, that is,
$\{u_n\}$ has a convergent subsequence whenever it satisfies
$\varphi_k'(u_n)\to 0$ as $n\to+\infty$ and
$\{\varphi_k(u_n)\}$ is bounded.
\end{lemma}
\begin{proof}
It follows from (H1) and Sobolev's inequality that
\begin{align}
&\Big|\int_0^{kT}(\nabla F(t,u_n(t)),\tilde{u}_n(t))dt\Big| \nonumber\\
&\leq \int_0^{kT}f(t)h(|\bar{u}_n+\tilde{u}_n(t)|)|\tilde{u}_n(t)|dt
+\int_0^{kT}g(t)|\tilde{u}_n(t)|dt \nonumber\\
&\leq \int_0^{kT}f(t)[C^*(h(|\bar{u}_n|)
+h(|\tilde{u}_n(t)|))]|\tilde{u}_n(t)|dt
+\|\tilde{u}_n\|_\infty\int_0^{kT}g(t)dt \nonumber \\
&\leq C^*[h(|\bar{u}_n|)+h(|\tilde{u}_n(t)|)]\|\tilde{u}_n\|_\infty
\int_0^{kT}f(t)dt
+\|\tilde{u}_n\|_\infty\int_0^{kT}g(t)dt \nonumber\\
&\leq C^*\Big[\frac{1}{2pC^*C_0^p}\|\tilde{u}_n\|_\infty^p
+2pC^*C_0^ph^q(|\bar{u}_n|)\Big(\int_0^{kT}f(t)dt\Big)^q\Big]
+\|\tilde{u}_n\|_\infty\int_0^{kT}g(t)dt \nonumber\\
&\quad +C^*h(\|\tilde{u}_n\|_\infty)\|\tilde{u}_n\|_\infty
\int_0^{kT}f(t)dt \label{3.1}\\
&\leq \frac{1}{2p}\int_0^{kT}|\dot{u}_n(t)|^pdt+C_2h^q(|\bar{u}_n|)
+C^*[K_1\|\tilde{u}_n\|_\infty^\alpha+K_2]\|\tilde{u}_n\|_\infty
\int_0^{kT}f(t)dt \nonumber\\
&\quad +\|\tilde{u}_n\|_\infty\int_0^{kT}g(t)dt \nonumber \\
&\leq \frac{1}{2p}\int_0^{kT}|\dot{u}_n(t)|^pdt+C_2h^q(|\bar{u}_n|)
+C_3\Big(\int_0^{kT}|\dot{u}_n(t)|^pdt\Big)^{(\alpha+1)/p} \nonumber\\
&\quad +C_4\Big(\int_0^{kT}|\dot{u}_n(t)|^pdt\Big)^{1/p}. \nonumber
\end{align}
Hence, we see that
\begin{align}
\|\tilde{u}_n\|_\infty
&\geq|(\varphi'_k(u_n),\tilde{u}_n)| \nonumber \\
&=\big|\int_0^{kT}|\dot{u}_n(t)|^pdt-\int_0^{kT}(\nabla
F(t,u_n(t)),\tilde{u}_n(t))dt\big| \label{3.2}\\
&\geq\big(1-\frac{1}{2p}\big)\int_0^{kT}|\dot{u}_n(t)|^pdt-C_2h^q(|\bar{u}_n|)
-C_3\Big(\int_0^{kT}|\dot{u}_n(t)|^pdt\Big)^{(\alpha+1)/p} \nonumber\\
&\quad -C_4\Big(\int_0^{kT}|\dot{u}_n(t)|^pdt\Big)^{1/p} \nonumber
\end{align}
for large $n$. It follows from Wirtinger's inequality that
\[ %\label{3.2}
\|\tilde{u}_n\|_\infty\leq\left(C_0+1\right)^{1/p}\left(\int_0^{kT}|\dot{u}_n(t)|^pdt\right)^{1/p}
\]
for all $n$, thus we obtain
\begin{equation} \label{3.3}
C_5h^q(|\bar{u}_n|)\geq\int_0^{kT}|\dot{u}_n(t)|^pdt-C_6
\end{equation}
for all large $n$, which implies that
\begin{align*}
\|\tilde{u}_n\|_\infty
&\leq\Big(C_0\int_0^{kT}|\dot{u}_n(t)|^pdt\Big)^{1/p}\\
&\leq [C_0(C_5h^q(|\bar{u}_n|)+C_6)]^{1/p}\\
&\leq [C_7(|\bar{u}_n|^{q\alpha}+1)]^{1/p}.
\end{align*}
Then one has
\begin{equation}\label{3.4}
|u_n(t)|
\geq|\bar{u}_n|-|\tilde{u}_n(t)|
\geq|\bar{u}_n|-\|\tilde{u}_n(t)\|_\infty
\geq|\bar{u}_n|-[C_7(|\bar{u}_n|^{q\alpha}+1)]^{1/p}
\end{equation}
for all large $n$ and every $t\in[0,kT]$.
We claim that $\{|\bar{u}_n|\}$ is bounded. Arguing indirectly, if
$\{|\bar{u}_n|\}$ is unbounded, we may assume that, going to a
subsequence if necessary,
\begin{equation} \label{3.5}
|\bar{u}_n|\to+\infty\quad\text{as }n\to+\infty,
\end{equation}
which, together with \eqref{3.4}, implies
\begin{equation} \label{3.6}
|u_n(t)|\geq\frac{1}{2}|\bar{u}_n|.
\end{equation}
Then for all large $n$ and every $t\in[0,kT]$, we have
\begin{equation} \label{3.7}
h(|\bar{u}_n|)\leq h(2|u_n(t)|)\leq 2C^*h(|u_n(t)|).
\end{equation}
Set $\delta=\frac{1}{2}\operatorname{meas}(E)$. In virtue of (H3) and
Lemma \ref{lemma2.1}, there exists a subset $E_\delta$ of $E$ with
meas$(E\backslash E_\delta)\delta>0,
\end{equation}
and for every $\beta>0$, there exists $M\geq1$ such that
\begin{equation} \label{3.10}
\frac{1}{h^q(|x|)}F(t,x)\geq\beta
\end{equation}
for all $|x|\geq M$ and all $t\in E_\delta$. By \eqref{3.5} and
\eqref{3.6}, one has
$|u_n(t)|\geq M$
for all large $n$ and every $t\in[0,kT]$. It follows from
\eqref{3.3}, \eqref{3.7}, \eqref{3.9}, \eqref{3.10} that
\begin{align*}
\varphi_k(u_n)
&=\frac{1}{p}\int_0^{kT}|\dot{u}_n(t)|^pdt-\int_0^{kT}F(t,u_n(t))dt\\
&\leq\frac{1}{p}(C_5h^q(|\bar{u}_n|)+C_6)-\int_{[0,kT]\backslash
E_\delta}\gamma(t)dt-\int_{E_\delta}\beta h^q(|u_n(t)|)dt\\
&\leq C_8h^q(|\bar{u}_n|)+C_9-\beta\int_{E_\delta}
\big(\frac{1}{2C^*}h(|\bar{u}_n|)\big)^qdt\\
&\leq C_8h^q(|\bar{u}_n|)+C_9-\beta\frac{1}{(2C^*)^q}h^q(|\bar{u}_n|)
\delta
\end{align*}
for all large $n$. So, we obtain
\[
\limsup_{n\to+\infty}\frac{1}{h^q(|\bar{u}_n|)}\varphi_k(u_n)\leq
C_8-\beta\frac{1}{(2C^*)^q}\delta.
\]
By the arbitrariness of $\beta>0$, one has
\[
\limsup_{n\to+\infty}\frac{1}{h^q(|\bar{u}_n|)}\varphi_k(u_n)=-\infty,
\]
which contradicts the boundedness of $\varphi_k(u_n)$. Hence
$\{|\bar{u}_n|\}$ is bounded. Furthermore, by \eqref{3.2} and
\eqref{3.3}, we know $\{u_n\}$ is bounded. Arguing then as in
\cite[Proposition 4.1]{Mawhin00}, we conclude that (PS) condition is
satisfied.
\end{proof}
\begin{lemma} \label{lemma3.2}
Assume that $F$ satisfies assumption {\rm (A1), (H2), (H4), (H5)}.
Then $\varphi_k$ satisfies the {\rm (PS)} condition.
\end{lemma}
\begin{proof}
It follows from \eqref{3.1} and Lemma \ref{lemma2.2}
that
\begin{align}
&\big|\int_0^{kT}(\nabla
F(t,u_n(t)),\tilde{u}_n(t))dt\big| \nonumber\\
&\leq C^*\Big[\frac{1}{2pC^*C_0^p}\|\tilde{u}_n\|_\infty^p
+2pC^*C_0^p{h^*}^q(|\bar{u}_n|)\Big(\int_0^{kT}f(t)dt\Big)^q\Big]
+\|\tilde{u}_n\|_\infty\int_0^{kT}g(t)dt
\nonumber\\
&\quad +C^*h^*(\|\tilde{u}_n\|_\infty)\|\tilde{u}_n\|_\infty
\int_0^{kT}f(t)dt \nonumber\\
&\leq \frac{1}{2p}\int_0^{kT}|\dot{u}_n(t)|^pdt
+C_2{h^*}^q(|\bar{u}_n|)
+C^*[\epsilon\|\tilde{u}_n\|_\infty^{p-1}
+C_1]\|\tilde{u}_n\|_\infty\int_0^{kT}f(t)dt \nonumber\\
&\quad +\|\tilde{u}_n\|_\infty\int_0^{kT}g(t)dt \label{3.11}\\
&\leq \big(\frac{1}{2p}+\epsilon C_{10}\big)
\int_0^{kT}|\dot{u}_n(t)|^pdt+C_2{h^*}^q(|\bar{u}_n|)
+C_{11}\Big(\int_0^{kT}|\dot{u}_n(t)|^pdt\Big)^{1/p}, \nonumber
\end{align}
which implies
\begin{align*}
\|\tilde{u}_n\|_\infty &\geq|(\varphi'_k(u_n),\tilde{u}_n)|\\
&\geq\big(1-\frac{1}{2p}-\epsilon
C_{10}\big)\int_0^{kT}|\dot{u}_n(t)|^pdt-C_2{h^*}^q(|\bar{u}_n|)
-C_{11}\Big(\int_0^{kT}|\dot{u}_n(t)|^pdt\Big)^{1/p}
\end{align*}
for large $n$. Thus, by \eqref{3.2}, one has
\begin{equation} \label{3.12}
C_{12}{h^*}^q(|\bar{u}_n|)\geq\int_0^{kT}|\dot{u}_n(t)|^pdt-C_{13}
\end{equation}
for all large $n$ and $\epsilon$ small enough, which implies
\[
\|\tilde{u}_n\|_\infty
\leq [C_0(C_{12}{h^*}^q(|\bar{u}_n|)+C_{13})]^{1/p}
\leq [C_{14}(\epsilon|\bar{u}_n|^p+1)]^{1/p}
\]
by Lemma \ref{lemma2.2}. Consequently, we get
\begin{equation}\label{3.13}
|u_n(t)|\geq|\bar{u}_n|-[C_{14}(\epsilon|\bar{u}_n|^p+1)]^{1/p}
\end{equation}
for all large $n$, every $t\in[0,kT]$ and $\epsilon$ small enough.
Assume $\{|\bar{u}_n|\}$ is unbounded, by \eqref{3.13}, for
$\epsilon$ small enough, we obtain
\begin{equation} \label{3.14}
|u_n(t)|\geq\frac{1}{2}|\bar{u}_n|.
\end{equation}
Combine \eqref{3.14} with $(H_4)$, one has
\[
H^*(|\bar{u}_n|)\leq 2C^*H^*(|u_n(t)|).
\]
With the same arguments of \eqref{3.8}-\eqref{3.10}, by (H5), we
know
\begin{equation}\label{3.15}
\frac{1}{H^*(|x|)}F(t,x)\geq C_{15}
\end{equation}
for all $|x|\geq M$ and all $t\in E_\delta$. We see that, jointly
with \eqref{3.12}, \eqref{3.15} and Lemma \ref{lemma2.2}, for all
large $n$,
\begin{align*}
\varphi_k(u_n)
&=\frac{1}{p}\int_0^{kT}|\dot{u}_n(t)|^pdt-\int_0^{kT}F(t,u_n(t))dt\\
&\leq\frac{1}{p}(C_{12}{h^*}^q(|\bar{u}_n|)+C_{13})
-\int_{[0,kT]\backslash
E_\delta}\gamma(t)dt-\int_{E_\delta}C_{15} H^*(|u_n(t)|)dt\\
&\leq C_{16}{h^*}^q(|\bar{u}_n|)+C_{17}
-C_{15}\int_{E_\delta}\frac{1}{2C^*}H^*(|\bar{u}_n|)dt\\
&\leq C_{16}{h^*}^q(|\bar{u}_n|)+C_{17}-C_{18}\delta H^*(|\bar{u}_n|),
\end{align*}
which implies
\begin{align*}
0&=\limsup_{n\to+\infty}\frac{1}{H^*(|\bar{u}_n|)}\varphi_k(u_n)\\
&\leq\limsup_{n\to+\infty}
\big[C_{16}\frac{{h^*}^q(|\bar{u}_n|)}{H^*(|\bar{u}_n|)}
+\frac{C_{17}}{H^*(|\bar{u}_n|)}-C_{18}\delta\big]
\leq-C_{18}\delta,
\end{align*}
a contradiction. Hence $\{|\bar{u}_n|\}$ is bounded, moreover, we
can get $\{u_n\}$ is bounded. So (PS) condition is satisfied, which
completes the proof.
\end{proof}
Now, we are ready to proof our main results.
\begin{proof}[Proof of Theorem \ref{thm1.1}]
It follows from
Lemma \ref{lemma3.1} that $\varphi_k$ satisfies the (PS) condition.
In order to use the saddle point theorem, we only need to verify the
following conditions
\begin{itemize}
\item[(I1)] $\varphi_k(u)\to+\infty$ as
$\|u\|\to+\infty$ in $\tilde{W}_{kT}^{1,p}$,
\item[(I2)] $\varphi_k(x+e_k(t))\to-\infty$ as
$|x|\to+\infty$ in $\mathbb{R^N}$,
\end{itemize}
where $\tilde{W}_{kT}^{1,p}:=\{u\in W_{kT}^{1,p}:\bar{u}=0\}$,
$e_k(t)=k\cos(k^{-1}\omega t)x_0\in\tilde{W}_{kT}^{1,p}$,
$x_0\in\mathbb{R^N}$, $|x_0|=1$ and
$\omega=\frac{2\pi}{T}$. Next, we show that $\varphi_k$ satisfies
(I1) and (I2).
For all $x\in\mathbb{R^N}$, it follows from \eqref{3.10} that
\begin{equation} \label{3.16}
\begin{split}
\varphi_k(x+e_k(t))
&=\frac{1}{p}\int_0^{kT}|\dot{e}_k(t)|^pdt
-\int_0^{kT}F(t,x+k\cos(k^{-1}\omega t)x_0)dt\\
&\leq\frac{1}{p}\int_0^{kT}|\omega(\sin k^{-1}\omega
t)x_0|^pdt-\int_{[0,kT]\backslash
E_\delta}\gamma(t)dt\\
&\quad -\beta\int_{E_\delta}h^q(|x+k\cos(k^{-1}\omega t)x_0|)dt\\
&\leq C_{19}k-\int_{[0,kT]\backslash
E_\delta}\gamma(t)dt-\beta h^q(M)\operatorname{meas}(E_\delta)
\end{split}
\end{equation}
for all $|x|\geq M+k$. By the arbitrariness of $\beta$, one has
\[
\varphi_k(x+e_k(t))\to-\infty \quad\text{as
}|x|\to+\infty\text{ in }\mathbb{R^N}.
\]
Thus (I2) is satisfied.
For all $u\in\tilde{W}_{kT}^{1,p}$, it follows from Sobolev's
inequality that
\begin{align*}
\big|&\int_0^{kT}[F(t,u(t))-F(t,0)]dt\big|
=\big|\int_0^{kT}(\nabla F(t,su(t)),u(t))dsdt\big|\\
&\leq\int_0^{kT}\int_0^1f(t)h(|su(t)|)|u(t)|dsdt
+\int_0^{kT}\int_0^1g(t)|u(t)|dsdt\\
&\leq\int_0^{kT}f(t)[K_1|u(t)|^\alpha+K_2]|u(t)|dt
+\int_0^{kT}g(t)|u(t)|dt\\
&\leq K_1\|u\|_\infty^{\alpha+1}\int_0^{kT}f(t)dt
+K_2|u\|_\infty\int_0^{kT}f(t)dt+\|u\|_\infty\int_0^{kT}g(t)dt\\
&\leq C_{20}\Big(\int_0^{kT}|\dot{u}(t)|^pdt\Big)^{(\alpha+1)/p}
+C_{21}\Big(\int_0^{kT}|\dot{u}(t)|^pdt\Big)^{1/p}.
\end{align*}
Hence, we have
\begin{align*}
\varphi_k(u)
&=\frac{1}{p}\int_0^{kT}|\dot{u}(t)|^pdt
-\int_0^{kT}[F(t,u(t))-F(t,0)]dt-\int_0^{kT}F(t,0)dt\\
&\geq \frac{1}{p}\int_0^{kT}|\dot{u}(t)|^pdt-C_{20}
\Big(\int_0^{kT}|\dot{u}(t)|^pdt\Big)^{(\alpha+1)/p}\\
&\quad -C_{21}\Big(\int_0^{kT}|\dot{u}(t)|^pdt\Big)^{1/p}-C_{22},
\end{align*}
then we can conclude that $\varphi_k(u)\to+\infty$
as $\|u\|\to+\infty$ in $\tilde{W}_{kT}^{1,p}$. Plainly,
condition (I1) holds.
By (I1), (I2) and the saddle point theorem, there exists a
critical point $u_k\in W_{kT}^{1,p}$ for $\varphi_k$ such that
\[
-\infty-\infty,
\]
which contradicts \eqref{3.18}. This completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm1.2}]
By Lemma \ref{lemma3.2}, $\varphi_k$ satisfies the (PS) condition.
By the argument of Theorem \ref{thm1.1}, we only need to check
that $\varphi_k$ satisfies (I1) and (I2).
In fact from \eqref{3.15}, for all $x\in\mathbb{R^N}$ it follows
that
\begin{align*}
\varphi_k(x+e_k(t))&=\frac{1}{p}\int_0^{kT}|\dot{e}_k(t)|^pdt-\int_0^{kT}F(t,x+k\cos(k^{-1}\omega
t)x_0)dt\\
&\leq C_{19}k-\int_{[0,kT]\backslash
E_\delta}\gamma(t)dt-C_{15}\int_{E_\delta}H^*(|x+k\cos(k^{-1}\omega
t)x_0|)dt
\end{align*}
for all $|x|\geq M+k$. Using the fact $H^*(t)\to+\infty$ as
$t\to+\infty$ of Lemma \ref{lemma2.2} and \eqref{3.9}, one
has
\[
\varphi_k(x+e_k(t))\to-\infty \quad\text{as }|x|\to+\infty
\text{ in }\mathbb{R^N}.
\]
Thus (I2) is safisfied.
For all $u\in\tilde{W}_{kT}^{1,p}$, we have
\begin{align*}
&\big|\int_0^{kT}[F(t,u(t))-F(t,0)]dt\big|\\
&\leq\int_0^{kT}\int_0^1f(t)h^*(|su(t)|)|u(t)|dsdt+\|u\|_\infty\int_0^{kT}g(t)dt\\
&\leq\int_0^{kT}f(t)[\epsilon|u(t)|^{p-1}+C_1]|u(t)|dt+\|u\|_\infty\int_0^{kT}g(t)dt\\
&\leq
\epsilon
C_{24}\int_0^{kT}|\dot{u}(t)|^pdt+C_{25}
\Big(\int_0^{kT}|\dot{u}(t)|^pdt\Big)^{1/p},
\end{align*}
which implies
\[
\varphi_k(u)\geq\big(\frac{1}{p}-\epsilon
C_{24}\big)\int_0^{kT}|\dot{u}(t)|^pdt
-C_{25}\Big(\int_0^{kT}|\dot{u}(t)|^pdt\Big)^{1/p}-C_{22}.
\]
Then for any $\epsilon$ small enough, we have
$\varphi_k(u)\to+\infty$ as $\|u\|\to+\infty$
in $\tilde{W}_{kT}^{1,p}$. So condition (I1) holds, and
the proof hereby is complete.
\end{proof}
\section{Example}
In this section, we give a new example to illustrate our results.
Consider function
\[
F(t,x)=|\sin\omega t||x|^{1+\alpha},
\]
where $0